Second-Order Logic over Finite Structures - Report on a Research Programme
نویسنده
چکیده
This talk will report on the results achieved so far in the context of a research programme at the cutting point of logic, formal language theory, and complexity theory. The aim of this research programme is to classify the complexity of evaluating formulas from different prefix classes of second-order logic over different types of finite structures, such as strings, graphs, or arbitrary structures. In particular, we report on classifications of second-order logic over graphs. Fagin’s theorem, the first important result of descriptive complexity, asserts that a property of graphs is in NP if and only if it is definable by an existential second-order formula. We studied the complexity of evaluating existential second-order formulas that belong to prefix classses of existential second-order logic, where a prefix class is the collection of all existential second-order formulas in prenex normal form such that the second-order and the first-order quantifiers obey a certain quantifier pattern. We completely characterize the computational complexity of prefix classes of existential second-order logic in three different contexts: (1) over directed graphs, (2) over undirected graphs with self-loops and (3) over undirected graphs without self-loops. We also obtained a complete classification of the regular and non-regular fragments of full second order logic over strings. The main results have been published in the following papers: Georg Gottlob, Phokion G. Kolaitis, Thomas Schwentick: Existential second-order logic over graphs: Charting the tractability frontier. J. ACM 51(2): 312-362 (2004) Thomas Eiter, Yuri Gurevich, Georg Gottlob: Existential second-order logic over strings. J. ACM 47(1): 77-131 (2000) -Thomas Eiter, Georg Gottlob, Thomas Schwentick: Second-Order Logic over Strings: Regular and Non-regular Fragments. Developments in Language Theory 2001: 37-56
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